A path to quantum gravity?

Joe Fitzsimons, Jonathan Jones and Vlatko Vedral just put out a fascinating and brilliant paper. I’ll be honest (no personal offense meant! 😀 ) and say that it is very much in the present style of writing physics and mathematics papers, which is to say as non-pedantic and jargon-laden as possible. This is a style I have come to dislike (think about this: we have freshman, non-science majors read Einstein’s original SR paper at Saint Anselm), but that is another story entirely.

Anyway, after some discussion with Joe about it I now see what they are getting at – and I think it offers a very intriguing potential path to follow in search of quantum gravity. It also seems to support my suspicion that time and space (or at least their connection via the metric) is emergent.

So here’s the basic argument. In quantum mechanics, density matrices are used to define quantum states that, in theory (depending on one’s interpretation), can extend throughout space. In other words, we use such matrices to represent what are usually spacelike separated measurement events. Joe, Jonathan, and Vlatko (henceforth FJV) ask if it might be possible to to extend these matrices in such a way as to cover a spread of time as well. In doing so they introduce the idea of a pseudo-density matrix (PDM). The PDM, like the usual density matrix, is positive semi-definite for spacelike separated events while it fails to be so for time like separated events (they don’t mention lightlike separations – in theory it should also fail to be positive semi-definite in this case). Intriguingly they actually present results of two qubit NMR experiments that support the results.

What really struck me is that they appear to be on the verge of being able to obtain the metric tensor of Minkowski space. In fact, Joe told me he had already recovered the signature which, given the results concerning the PDM, tells me they’ve already got the metric. In the signature convention (+,+,+,-), the metric tensor is indeed positive semi-definite for spacelike separations and not for timelike and lightlike separations.

The only thing I caution is that, though we like to proclaim the contrary (and FJV say it in their abstract), time and space are not the same in special relativity. If they were, time would have the same sign as space in the metric tensor (incidentally, Scott Aaronson agrees with me on this). In addition, if time and space were truly equal we would be able to go back in time.

Nevertheless, I think this is one of the most promising potential routes to quantum gravity that I’ve seen in years. It brings the “language” of quantum mechanics and relativity just a bit closer (something I’ve long thought about, but more in group theory terms) and it hints at the emergence of time (and maybe even space). It will be interesting to see where this leads.

2 Responses to “A path to quantum gravity?”

Let me add a comment to try to clarify one or two points. In our construction it is not guaranteed that all systems which are time-like separated will necessarily lead to negative eigenvalues, and hence it doesn’t immediately recover the full structure of space-time. I would also point out that our construction has no notion of distances between points, so in its current form, the signature is the most you can hope for, rather than the metric tensor itself. I think that you might be able to get the full metric tensor if you fix the Hamiltonian in some frame, but that is still an open question.

On the other hand, there are some nice features which might be useful, such as the fact that any spacelike cut should result in a valid density matrix and should not necessarily preserve particle number.

Regarding light-like separation, it should be straight forward to see that in this case it is possible to get a maximal value for our measure of causality. One way to see this is to consider the example of two sequential measurement events on a single particle which we discuss in the paper. If this particle were a photon, these events would be light-like separated.